Automorphisms of partition order-decreasing transformation monoids.
type: Item
instance of: scholarly article
title: Automorphisms of partition order-decreasing transformation monoids.
zbMATH Open document ID: 1266.20077
author: Haobo Yang
author: Xiuliang Yang
published in: Semigroup Forum
publication date: 2013-03-04T00:00:00Z
review text: Let \(T_n\) (\(S_n\)) be the full transformation semigroup (the symmetric group, respectively) on an \(n\)-element set \(X_n\), \(\rho\) an equivalence relation on \(X_n\), \(\preceq\) a total order on \(X_n/\rho\), \(T(\rho,\preceq)=\{\alpha\in T_n:(x\alpha)\rho\preceq x\rho,\;\forall x\in X_n\}\) and \(U_\rho=\{\mu\in S_n:(x\rho)\mu=x\rho,\;x\in X_n\}\). The main theorem states that \(U_\rho\) is the group of units of \(T(\rho,\preceq)\) and for any \(\varphi\in\Aut\,T(\rho,\preceq)\), there exists \(\mu\in U_\rho\) such that \((\alpha)\varphi=\mu^{-1}\alpha\mu\) for all \(\alpha\in T(\rho,\preceq)\). Conversely, if \(\mu\in U_\rho\), then \(\mu^{-1}\alpha\mu\in T(\rho,\preceq)\) for any \(\alpha\in T(\rho,\preceq)\), and the map \(\varphi\colon T(\rho,\preceq)\to T(\rho,\preceq)\) defined by \((\alpha)\varphi=\mu^{-1}\alpha\mu\), \(\alpha\in T(\rho,\preceq)\), is an automorphism.
reviewed by: Peeter Normak
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20M15
Mathematics Subject Classification ID: 20B25
zbMATH DE Number: 6140995
zbMATH Keywords: transformation semigroups
zbMATH Keywords: partition order-decreasingg transformations
zbMATH Keywords: groups of units
zbMATH Keywords: automorphism groups
MaRDI profile type: Publication
full work available at URL: s00233-012-9430-2
OpenAlex ID: W2032791895
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cites work: Automorphisms of transformation semigroups
cites work: On the semigroups of order-decreasing finite full transformations
DOI: 10.1007/S00233-012-9430-2
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description: scientific article; zbMATH DE number 6140995